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Exploring the structure and function of temporal networks with dynamic graphlets

Yuriy Hulovatyy, Huili Chen, Tijana Milenkovic

arXiv:1412.3885v1cs.SIphysics.soc-phq-bio.MN

TL;DR

Temporal-network analysis must account for information that aggregation and independent snapshots lose, while existing temporal motif approaches have constraints including null-model dependence. The paper develops dynamic graphlets for temporal structure and reports that they outperform static and static-temporal graphlets across network and node characterization tasks. It concludes that accounting for temporal information helps characterize network structure and function.

  • Problem

    Aggregate and snapshot-based approaches discard temporal information, while temporal motif methods have limitations including reliance on null models.

    Source: A. Motivation · p. 1, B. Related work · p. 2

  • Method

    The paper develops dynamic graphlets to represent temporal subgraphs and their event order, and presents an approach for enumerating and counting them.

    Source: C. Our contribution · p. 2, A. Dynamic graphlets · p. 3

  • Results

    Dynamic graphlets outperform static and static-temporal graphlet approaches in tasks characterizing the structure and function of temporal networks and individual nodes.

    Source: C. Our contribution · p. 3

  • Takeaways & Limitations

    The paper concludes that accounting for temporal information helps with structural and functional interpretation of network data.

    Source: §IV · p. 10

    Dense neighborhoods with many events between the same node pairs can produce many dynamic graphlets, including artifacts when consecutive snapshots share a dense structure.

    Source: C. Causal counting of dynamic… · p. 5

Abstract

from arXiv · show

With the growing amount of available temporal real-world network data, an important question is how to efficiently study these data. One can simply model a temporal network as either a single aggregate static network, or as a series of time-specific snapshots, each of which is an aggregate static network over the corresponding time window. The advantage of modeling the temporal data in these two ways is that one can use existing well established methods for static network analysis to study the resulting aggregate network(s). Here, we develop a novel approach for studying temporal network data more explicitly. We base our methodology on the well established notion of graphlets (subgraphs), which have been successfully used in numerous contexts in static network research. Here, we take the notion of static graphlets to the next level and develop new theory needed to allow for graphlet-based analysis of temporal networks. Our new notion of dynamic graphlets is quite different than existing approaches for dynamic network analysis that are based on temporal motifs (statistically significant subgraphs). Namely, these approaches suffer from many limitations. For example, they can only deal with subgraph structures of limited complexity. Also, their major drawback is that their results heavily depend on the choice of a null network model that is required to evaluate the significance of a subgraph. However, choosing an appropriate null network model is a non-trivial task. Our dynamic graphlet approach overcomes the limitations of the existing temporal motif-based approaches. At the same time, when we thoroughly evaluate the ability of our new approach to characterize the structure and function of an entire temporal network or of individual nodes, we find that the dynamic graphlet approach outperforms the static graphlet approach, which indicates that accounting for temporal information helps.

I. INTRODUCTION

Temporal networks provide time-stamped changes in systems that static-network methods cannot directly analyze. Aggregation and snapshots make static methods usable but lose temporal information, motivating methods that account for changes across time.

A. Motivation

  • Aggregating a temporal network into one static graph loses temporal information, while analyzing snapshots independently misses relationships between snapshots.The paper argues that both approaches overlook information important for studying system evolution.Source: A. Motivation · p. 1

B. Related work

  • Static graphlets are induced subgraphs used to capture detailed network topology without the null model required by network motifs.Graphlets have supported measures of network and node similarity, as well as applications including alignment and clustering.Source: B. Related work · p. 2
  • Temporal motif methods extend local analysis across time, but some are limited to small or simple structures, such as linear paths.These restrictions limit their ability to capture complex network structure in detail.Source: B. Related work · p. 2
  • Temporal motif methods can constrain how many events a node participates in at a time and may provide network-level, but not individual-node, signatures.The latter limitation matters for linking a node’s topological position to function, for example through alignment or clustering.Source: B. Related work · p. 2
  • Temporal motif methods rely on null models, whose appropriate selection is especially complex in dynamic settings; they are also typically designed for directed networks.The paper notes that this makes them unsuitable for direct application to many undirected networks.Source: B. Related work · p. 2
  • An earlier static-temporal graphlet approach counted graphlets within snapshots but ignored relationships between snapshots.It was used to study human aging in protein-protein interaction networks, while the paper identifies inter-snapshot information as still unexplored.Source: B. Related work · p. 2

C. Our contribution

The paper develops dynamic graphlets to represent temporal network structure, including relationships between snapshots. It evaluates their use for network and node characterization and reports that they outperform static and static-temporal graphlet approaches in those tasks.

C. Our contribution

  • Dynamic graphlets support structures of arbitrary complexity, unrestricted node event participation, whole-network and individual-node signatures, and analysis without a null model.They also complement temporal motif approaches by working with undirected networks and explicitly represent relationships between snapshots.Source: C. Our contribution · p. 2
  • The illustrated static graphlets have up to four nodes and 15 node symmetry groups, with different node orbits shown in different colors.Nodes in the same orbit are topologically identical within that graphlet.Source: Figure 1 · p. 3
  • The comparison with static and static-temporal graphlets tests how accuracy varies with the amount of temporal information each approach considers.Static graphlets ignore temporal information, while static-temporal graphlets ignore inter-snapshot relationships.Source: C. Our contribution · pp. 2–3
  • Dynamic graphlets outperform static and static-temporal graphlet approaches in characterizing the structure and function of entire temporal networks and individual nodes.The evaluation covers synthetic and real-world temporal data and tasks involving grouping similar and separating dissimilar networks or nodes.Source: C. Our contribution · p. 3

II. METHODS

The methods section introduces dynamic graphlets, a counting algorithm, a related causal variant, and an experimental setup.

II. METHODS

  • The methods section introduces the theoretical notion of dynamic graphlets.It also presents an algorithm for counting them.Source: §II · p. 3
  • The methods section discusses causal dynamic graphlets as a related notion.Source: §II · p. 3
  • The methods section describes the experimental setup and evaluation framework.Source: §II · p. 3

A. Dynamic graphlets

Dynamic graphlets represent temporal subgraphs through their structure and the relative order of events, using a stricter causal connectivity condition. The section defines their node orbits and static backbones, then gives a recursive way to count and generate them.

A. Dynamic graphlets

  • A dynamic graphlet is an equivalence class of isomorphic ∆t-causal temporal subgraphs, distinguished by the relative order of events.A ∆t-causal subgraph is ∆t-connected, but the converse does not always hold.Source: A. Dynamic graphlets · p. 3
  • Removing event order and duplicate events on an edge gives each dynamic graphlet one static backbone, while a backbone can represent multiple dynamic graphlets.The backbones are the corresponding static graphlets.Source: A. Dynamic graphlets · p. 3, Figure 3 · p. 4
  • For a dynamic graphlet with n nodes and k events, node orbits reflect both topological and temporal positions; when n > 2, it has n different orbits.For n = 2, the undirected event endpoints are topologically equivalent, so there is one orbit.Source: Figure 2 · p. 3, A. Dynamic graphlets · p. 4
  • For n≥3 and k≥n−1, each k-event dynamic graphlet has a unique (k−1)-event prefix and is formed by adding an event between existing nodes or between an existing and a new node.For n−1=2, only one extension yields a three-node graphlet because the two nodes are indistinguishable.Source: A. Dynamic graphlets · p. 4
  • The recurrence is D(3, k) = 3D(3, k −1) + D(2, k −1) for n = 3, and D(n, k) = (2n −3)D(n, k −1) + 2D(n −1, k −1) for n > 3.The two-node case has D(2, k) = 1 for any k.Source: A. Dynamic graphlets · p. 4
  • Supplementary Table S2 reports dynamic graphlet counts for up to n = 11 nodes and k = 10 events.The authors obtain a closed-form solution by expanding the recurrence for small n and k.Source: A. Dynamic graphlets · p. 4
  • Enumeration starts from the single-event graphlet D0 and recursively extends it until the desired size is reached.Each dynamic graphlet has a unique preceding prefix.Source: A. Dynamic graphlets · p. 4

B. Counting dynamic graphlets in a network

The counting procedure grows graphlets recursively from each event in the temporal network. Its running time depends on network structure and is exponential in graphlet size.

B. Counting dynamic graphlets in a network

  • For each temporal-network event, counting starts with D0 and searches recursively for larger graphlets grown from that event.The procedure follows the graphlet-generation approach.Source: B. Counting dynamic graphlets in a… · p. 4
  • D9 has seven possible four-event extensions: five retain four nodes, while two add a node.The newest event must involve at least one node from event 3.Source: Figure 4 · p. 5
  • Counting all dynamic graphlets with up to k events takes O(|E| + |E|(D/|E|)^(k−1)) and is exponential in graphlet size.Here D is the number of ∆t-adjacent event pairs; running time depends on network structure.Source: B. Counting dynamic graphlets in a… · p. 5

C. Causal counting of dynamic graphlets in a network

Dense neighborhoods with repeated interactions can produce many dynamic graphlets that are likely artifacts. Causal counting adds a restriction for extensions across different edges to exclude some of these candidates and is expected to reduce running time.

C. Causal counting of dynamic graphlets in a network

  • Dense neighborhoods with many events between the same node pairs can yield numerous dynamic graphlets, including artifacts of repeated snapshot structure.The authors expect redundant counts to be reduced by modifying the counting process.Source: C. Causal counting of dynamic… · p. 5
  • For extensions across different edges, the new event’s endpoints cannot interact between the preceding event’s start and the new event’s start.This adds a third condition to the regular counting procedure.Source: C. Causal counting of dynamic… · p. 5
  • In the illustrated three-snapshot network, regular counting finds D1, D2, and D9, while causal counting finds only D1 and D2.D9 is excluded because c and d interact in both the second and third snapshots, violating the third condition.Source: Figure 5 · p. 5
  • The added condition does not apply when consecutive events use the same edge, preserving graphlets with repeated events on that edge.Applying it in this case would prevent such graphlets from being counted.Source: C. Causal counting of dynamic… · p. 5
  • Causal dynamic graphlet counting examines fewer options by excluding graphlets likely to be repeated-event artifacts and is expected to be more computationally efficient.The efficiency claim is an expected consequence of considering fewer options.Source: C. Causal counting of dynamic… · pp. 5–6

D. Experimental setup

The experiments compare four graphlet approaches using aggregate or snapshot-based network representations, varying graphlet sizes and temporal parameters. Classification evaluates whether objects with matching labels are closer in graphlet-based PCA space, against randomized baselines.

D. Experimental setup

  • Four approaches are compared: static, static-temporal, dynamic, and causal dynamic graphlets, using aggregate networks for static counting and snapshots for the other approaches.Snapshot windows have size tw; w sets the event threshold for adding edges. The dynamic approaches vary both graphlet node count n and event count k.Source: D. Experimental setup · p. 6
  • Network classification uses nine synthetic network classes, while node classification tests seven user-role labels in the Enron email network.The synthetic classes arise from variations of a network evolution model; Enron contains email communications from 2000 to 2002.Source: D. Experimental setup · pp. 6–7
  • The reported settings are w = 1 and tw = 2, after testing alternatives that produced no qualitative differences in results.Other tested temporal windows for synthetic networks and Enron also yielded qualitatively similar results.Source: D. Experimental setup · pp. 6–7
  • For network classification, the first two PCA components explain over 90% of graphlet-vector variation; node classification keeps the first three components.Euclidean distances in the resulting PCA spaces measure similarity between networks or nodes.Source: D. Experimental setup · pp. 6–7
  • Performance is assessed by ordering object pairs from nearest to farthest, testing whether same-class pairs rank closer, and comparing correct and incorrect pair distances with a Wilcoxon rank-sum test.The evaluation also compares methods with simple random embeddings and label-permuted counterparts; each randomization averages 10 runs and reports the highest-scoring random result.Source: D. Experimental setup · p. 7
  • AUPR, AUROC, break-even point, and maximum F-score summarize classification performance across retrieved pairs.AUPR is treated as more credible than AUROC when same-class and different-class pair counts are imbalanced.Source: D. Experimental setup · p. 7

III. RESULTS AND DISCUSSION

The results section evaluates dynamic graphlets against static and static-temporal graphlets. It examines both network classification and node classification, as well as the effects of method parameters.

III. RESULTS AND DISCUSSION

  • The evaluation compares novel dynamic graphlets with existing static and static-temporal graphlet approaches.Source: §III · p. 7
  • The comparison covers network classification and node classification tasks.Source: §III · p. 7
  • The section also discusses how different method parameters affect results.Source: §III · p. 7

A. Network classification

Across nine synthetic temporal-network classes, dynamic graphlets—especially causal dynamic graphlets—outperform static-temporal and static graphlets in accuracy and computational complexity. Regular dynamic graphlets are more accurate than causal dynamic graphlets, whose computational complexity is comparable.

A. Network classification

  • Regular dynamic graphlets achieve higher accuracy than causal dynamic graphlets, while their computational complexity is comparable.Source: A. Network classification · p. 7
  • Table I reports classification measures and running times; its running-time illustration covers one of the nine network classes and one specified parameter setting.The remaining classes' running times are in Supplementary Table S3; static and static-temporal implementations count graphlets up to five nodes by default, then remove larger-graphlet counts for smaller sizes.Source: Table I · p. 8, A. Network classification · p. 8
  • (Causal) dynamic graphlets outperform static-temporal graphlets, which in turn outperform static graphlets, in accuracy and computational complexity across nine synthetic temporal-network classes.All methods show non-random behavior: intra-class distances are significantly below inter-class distances (Wilcoxon rank-sum p-values less than 10−100).Source: A. Network classification · p. 7, Figure 6 · p. 8, Table I · p. 8

B. Node classification

On the Enron node-classification task, (causal) dynamic graphlets outperform static-temporal and static graphlets in accuracy and computational complexity. Results also qualify the comparison between dynamic variants by graphlet size and reveal differences in which nodes methods identify as similar.

B. Node classification

  • Node classification could test graphlets on five or more nodes, unlike network classification, because it counts graphlets in one network rather than across multiple networks.Source: B. Node classification · p. 9
  • (Causal) dynamic graphlets outperform static-temporal graphlets, which outperform static graphlets, in accuracy and computational complexity across six node classes.All methods show non-random behavior, with intra-class distances significantly below inter-class distances (Wilcoxon rank-sum p-values less than 10−100).Source: B. Node classification · p. 8, Figure 8 · p. 9, Figure 9 · p. 9
  • Causal dynamic graphlets outperform regular dynamic graphlets even when comparing graphlets of the same size.The best parameter version of causal dynamic graphlets also performs better, but that comparison is qualified because causal counting allows six- or seven-node graphlets that regular dynamic counting cannot feasibly attain.Source: B. Node classification · pp. 8–9
  • Causal dynamic graphlet counting takes significantly less time than regular dynamic graphlet counting in node classification.Source: B. Node classification · p. 9, Table II · p. 9
  • Within dynamic and causal dynamic methods, node-similarity results form clusters by three-node graphlets with different event counts, suggesting node count separates methods more than event count.Figure 10 compares methods using Jaccard similarity between their top 5% closest node pairs in graphlet-based PCA space.Source: B. Node classification · p. 9, Figure 10 · p. 10

C. Effect of graphlet size on results

The effects of graphlet size depend on the classification task and graphlet method: larger graphlets can reduce network-classification accuracy but help some node classifiers. Increasing graphlet size also increases computational complexity.

C. Effect of graphlet size on results

IV. CONCLUSIONS

The conclusion presents dynamic graphlets as an explicit way to capture temporal information in networks. Evaluation found them superior to static counterparts for structural and functional interpretation.

IV. CONCLUSIONS

  • Dynamic graphlets explicitly capture temporal information, addressing limitations of aggregate or snapshot-based static representations.Source: §IV · p. 10
  • A systematic evaluation found the dynamic graphlet approach superior to static counterparts for characterizing network structure and function.Source: §IV · p. 10
  • The conclusion links accounting for temporal information with improved structural and functional interpretation, and notes the method’s relevance as temporal network data grow.Source: §IV · p. 10
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